Coordinate Proofs

Geometry · Axiom Academy

Proving geometric theorems with algebra — place a figure on the plane, then let distance, slope, and midpoint do the proving. 1. The Coordinate-Proof Recipe A coordinate proof turns a geometry claim into algebra. The whole game is where you put the figure : a smart placement makes the numbers fall out cleanly. Press play and watch a triangle settle onto the grid — one vertex snaps to the origin, one side drops flat onto the x-axis. Then the recipe appears below. Slope (parallel / perpendicular) · Midpoint (bisection / centers) A floating triangle rotates down and snaps : vertex → origin, base → x-axis. Position it strategically — use the origin and axes to simplify the calculations. Use variables like a , b for general proofs, or specific numbers for a concrete example. Distance, slope, midpoint — to find lengths, check parallel/perpendicular, locate centers. Show the computed values prove the statement. 2. Prove a Triangle Is Isosceles Claim. Triangle ABC with A(0,0) , B(4,0) , C(2,3) is isosceles. Play the animation: each slanted side draws with its rise/run legs, and the two lengths settle to the same value , . The written proof unfolds afterward. Sides AC and BC each build from a rise-of-3, run-of-2 right triangle → both measure . To prove isosceles, show that two sides have equal length. Since , triangle ABC has two equal sides. Therefore, triangle ABC is isosceles. ∎ 3. Prove the Diagonals of a Rectangle Bisect Each Other

This is the written version of the interactive lesson above. See the full Geometry course.